Theano-Deep Learning Tutorials 笔记：Modeling and generating sequences of polyphonic music with the RNN
教程地址：http://www.deeplearning.net/tutorial/rnnrbm.html#rnnrbm
代码，数据集，论文 见教程。
The RNN-RBM
RNN-RBM也是能量模型，用于对时间序列的密度估计，在 time step t 的特征向量 为高维向量。
它可以描述多峰的条件概率分布 , where。
表示在 time t 时刻，历史序列（t 之前所有）。
每一个 time step 就是一个 RBM，而RBM的参数
又由 RNN（隐藏层为）决定。
(1)
(2)
RNN隐藏层表示为：
(3)
最终模型图示：
The overall probability distribution is given by the sum over the time stepsin agiven sequence:
(4)
where the right-hand side multiplicand is the marginalized probability of the RBM.
Implementation
教程实现了两个函数：一个训练RNN-RBM，另一个生成采样序列。
训练时，有
, RNN隐藏层 and 参数
（可以计算出）。参数更新为SGD随机梯度下降，和RBM训练类似，使用 contrastive divergence （CD）算法。
序列的生成和RNN相似，只是
在每个time step 都需要按RBM中的Gibbs采样得出。
The RBM layer
函数 build_rbm 建立RBM部分（图例中上部）Gibbs链，输入为 mini-batch(a binary matrix)；输入也可以不是 mini-batch（也可以说当mini-batch为1时），a binary vector。
def build_rbm(v, W, bv, bh, k):
'''Construct a k-step Gibbs chain starting at v for an RBM.
v : Theano vector or matrix
If a matrix, multiple chains will be run in parallel (batch).
W : Theano matrix
Weight matrix of the RBM.
bv : Theano vector
Visible bias vector of the RBM.
bh : Theano vector
Hidden bias vector of the RBM.
k : scalar or Theano scalar
Length of the Gibbs chain.
Return a (v_sample, cost, monitor, updates) tuple:
v_sample : Theano vector or matrix with the same shape as `v`
Corresponds to the generated sample(s).
cost : Theano scalar
Expression whose gradient with respect to W, bv, bh is the CD-k
approximation to the log-likelihood of `v` (training example) under the
RBM. The cost is averaged in the batch case.
monitor: Theano scalar
Pseudo log-likelihood (also averaged in the batch case).
updates: dictionary of Theano variable -> Theano variable
The `updates` object returned by scan.'''
def gibbs_step(v):
mean_h = T.nnet.sigmoid(T.dot(v, W) + bh)
h = rng.binomial(size=mean_h.shape, n=1, p=mean_h,
dtype=theano.config.floatX)
mean_v = T.nnet.sigmoid(T.dot(h, W.T) + bv)
v = rng.binomial(size=mean_v.shape, n=1, p=mean_v,
dtype=theano.config.floatX)
return mean_v, v
chain, updates = theano.scan(lambda v: gibbs_step(v)[1], outputs_info=[v],
n_steps=k)
v_sample = chain[-1]
mean_v = gibbs_step(v_sample)[0]
monitor = T.xlogx.xlogy0(v, mean_v) + T.xlogx.xlogy0(1 - v, 1 - mean_v)
monitor = monitor.sum() / v.shape[0]
def free_energy(v):
return -(v * bv).sum() - T.log(1 + T.exp(T.dot(v, W) + bh)).sum()
cost = (free_energy(v) - free_energy(v_sample)) / v.shape[0]
return v_sample, cost, monitor, updates
The RNN layer
函数 build_rnnrbm 融合RNN和RBM，关系如上文图例。
RNN部分（图例中下部）在训练时： 已知，RNN的训练不需要RBM的参数，先把RNN中隐藏层 u0到uT 按公式（3）计算出来，再把 T 个RBM 一次性构建出来。
模型训练完成后：RNN和RBM互相影响， 需在每个 time step t 通过 t 时刻的 RBM Gibbs采样得到，从而计算 t 时刻的RNN隐藏层 u以及后续 time step 的RBM（公式（2，3））和RNN。
def build_rnnrbm(n_visible, n_hidden, n_hidden_recurrent):
'''Construct a symbolic RNN-RBM and initialize parameters.
n_visible : integer
Number of visible units.
n_hidden : integer
Number of hidden units of the conditional RBMs.
n_hidden_recurrent : integer
Number of hidden units of the RNN.
Return a (v, v_sample, cost, monitor, params, updates_train, v_t,
updates_generate) tuple:
v : Theano matrix
Symbolic variable holding an input sequence (used during training)
v_sample : Theano matrix
Symbolic variable holding the negative particles for CD log-likelihood
gradient estimation (used during training)
cost : Theano scalar
Expression whose gradient (considering v_sample constant) corresponds
to the LL gradient of the RNN-RBM (used during training)
monitor : Theano scalar
Frame-level pseudo-likelihood (useful for monitoring during training)
params : tuple of Theano shared variables
The parameters of the model to be optimized during training.
updates_train : dictionary of Theano variable -> Theano variable
Update object that should be passed to theano.function when compiling
the training function.
v_t : Theano matrix
Symbolic variable holding a generated sequence (used during sampling)
updates_generate : dictionary of Theano variable -> Theano variable
Update object that should be passed to theano.function when compiling
the generation function.'''
W = shared_normal(n_visible, n_hidden, 0.01)
bv = shared_zeros(n_visible)
bh = shared_zeros(n_hidden)
Wuh = shared_normal(n_hidden_recurrent, n_hidden, 0.0001)
Wuv = shared_normal(n_hidden_recurrent, n_visible, 0.0001)
Wvu = shared_normal(n_visible, n_hidden_recurrent, 0.0001)
Wuu = shared_normal(n_hidden_recurrent, n_hidden_recurrent, 0.0001)
bu = shared_zeros(n_hidden_recurrent)
params = W, bv, bh, Wuh, Wuv, Wvu, Wuu, bu # learned parameters as shared
# variables
v = T.matrix() # a training sequence
u0 = T.zeros((n_hidden_recurrent,)) # initial value for the RNN hidden
# units
# If `v_t` is given, deterministic recurrence to compute the variable
# biases bv_t, bh_t at each time step. If `v_t` is None, same recurrence
# but with a separate Gibbs chain at each time step to sample (generate)
# from the RNN-RBM. The resulting sample v_t is returned in order to be
# passed down to the sequence history.
def recurrence(v_t, u_tm1):
bv_t = bv + T.dot(u_tm1, Wuv)
bh_t = bh + T.dot(u_tm1, Wuh)
generate = v_t is None
if generate:
v_t, _, _, updates = build_rbm(T.zeros((n_visible,)), W, bv_t,
bh_t, k=25)
u_t = T.tanh(bu + T.dot(v_t, Wvu) + T.dot(u_tm1, Wuu))
return ([v_t, u_t], updates) if generate else [u_t, bv_t, bh_t]
# For training, the deterministic recurrence is used to compute all the
# {bv_t, bh_t, 1 <= t <= T} given v. Conditional RBMs can then be trained
# in batches using those parameters.
(u_t, bv_t, bh_t), updates_train = theano.scan(
lambda v_t, u_tm1, *_: recurrence(v_t, u_tm1),
sequences=v, outputs_info=[u0, None, None], non_sequences=params)
v_sample, cost, monitor, updates_rbm = build_rbm(v, W, bv_t[:], bh_t[:],
k=15)
updates_train.update(updates_rbm)
# symbolic loop for sequence generation
(v_t, u_t), updates_generate = theano.scan(
lambda u_tm1, *_: recurrence(None, u_tm1),
outputs_info=[None, u0], non_sequences=params, n_steps=200)
return (v, v_sample, cost, monitor, params, updates_train, v_t,
updates_generate)
Putting it all together
class RnnRbm:
'''Simple class to train an RNN-RBM from MIDI files and to generate sample
sequences.'''
def __init__(
self,
n_hidden=150,
n_hidden_recurrent=100,
lr=0.001,
r=(21, 109),
dt=0.3
):
'''Constructs and compiles Theano functions for training and sequence
generation.
n_hidden : integer
Number of hidden units of the conditional RBMs.
n_hidden_recurrent : integer
Number of hidden units of the RNN.
lr : float
Learning rate
r : (integer, integer) tuple
Specifies the pitch range of the piano-roll in MIDI note numbers,
including r[0] but not r[1], such that r[1]-r[0] is the number of
visible units of the RBM at a given time step. The default (21,
109) corresponds to the full range of piano (88 notes).
dt : float
Sampling period when converting the MIDI files into piano-rolls, or
equivalently the time difference between consecutive time steps.'''
self.r = r
self.dt = dt
(v, v_sample, cost, monitor, params, updates_train, v_t,
updates_generate) = build_rnnrbm(
r[1] - r[0],
n_hidden,
n_hidden_recurrent
)
gradient = T.grad(cost, params, consider_constant=[v_sample])
updates_train.update(
((p, p - lr * g) for p, g in zip(params, gradient))
)
self.train_function = theano.function(
[v],
monitor,
updates=updates_train
)
self.generate_function = theano.function(
[],
v_t,
updates=updates_generate
)
def train(self, files, batch_size=100, num_epochs=200):
'''Train the RNN-RBM via stochastic gradient descent (SGD) using MIDI
files converted to piano-rolls.
files : list of strings
List of MIDI files that will be loaded as piano-rolls for training.
batch_size : integer
Training sequences will be split into subsequences of at most this
size before applying the SGD updates.
num_epochs : integer
Number of epochs (pass over the training set) performed. The user
can safely interrupt training with Ctrl+C at any time.'''
assert len(files) > 0, 'Training set is empty!' \
' (did you download the data files?)'
dataset = [midiread(f, self.r,
self.dt).piano_roll.astype(theano.config.floatX)
for f in files]
try:
for epoch in range(num_epochs):
numpy.random.shuffle(dataset)
costs = []
for s, sequence in enumerate(dataset):
for i in range(0, len(sequence), batch_size):
cost = self.train_function(sequence[i:i + batch_size])
costs.append(cost)
print('Epoch %i/%i' % (epoch + 1, num_epochs))
print(numpy.mean(costs))
sys.stdout.flush()
except KeyboardInterrupt:
print('Interrupted by user.')
def generate(self, filename, show=True):
'''Generate a sample sequence, plot the resulting piano-roll and save
it as a MIDI file.
filename : string
A MIDI file will be created at this location.
show : boolean
If True, a piano-roll of the generated sequence will be shown.'''
piano_roll = self.generate_function()
midiwrite(filename, piano_roll, self.r, self.dt)
if show:
extent = (0, self.dt * len(piano_roll)) + self.r
pylab.figure()
pylab.imshow(piano_roll.T, origin='lower', aspect='auto',
interpolation='nearest', cmap=pylab.cm.gray_r,
extent=extent)
pylab.xlabel('time (s)')
pylab.ylabel('MIDI note number')
pylab.title('generated piano-roll')
Results
在Nottingham数据集上运行 200 epochs，训练大约 24 小时。
The figures below show the piano-rolls of two sample sequences and we provide the corresponding MIDI files:
Listen to sample1.mid
Listen tosample2.mid
感觉教程对数据集介绍不太清楚，不太知道输入输出是啥，下面介绍下：
不难看出 piano-rolls 为输入，为持续60s的序列（对应 time step）样本，黑色表示1，白色表示0，对应文中提到的输入为 binary ，看下图应该很容易知道 piano-rolls 具体是啥了。
Nottingham数据集中每个样本为一个行数为150左右（代表150个time step），列数为88（代表21-109的midi note number，对应钢琴的音域A0到C8，可以说音域吗。。。）的矩阵，代表一段曲子的 piano-rolls，矩阵元素都为0或1。
目的是通过 序列piano-rolls 预测 后续 piano-rolls。
MIDI note number 60 就是"Middle C"或C5。就是唱歌弹琴乐谱啥的里面那个音高的数值化，范围大概是0到127（钢琴是21-109），数值越大音调越高。